arXiv subjects
Publications and source records attributed to Ziemowit Kostana.
We study Fraïssé classes of Abelian groups with pure embeddings. We characterize Abelian groups that are universal and homogeneous for: finitely co-generated groups, finite groups, groups of size less that $κ$, where $κ$ is strongly inaccessible.
Given a strongly inaccessible cardinal $λ$, we study the Fraïssé class of all Boolean algebras of size $<λ$, together with regular embeddings. We prove that this is indeed a Fraïsséclass, and its limit has the same completion as the Lévy collapse. We also give a direct proof that the collapsing algebra of density $κ$ is not the union of a $κ$-chain of regular sub-algebras of density $<κ$.
We study countable graphs that -- up to isomorphism and with probability one -- arise from a random process, in a similar fashion as the Rado graph. Unlike in the classical case, we do not require that probabilities assigned to pairs of points are all equal. We give examples of such generalized random graphs, and show that the class of graphs under consideration has a two-element basis.
We introduce and study a family of axioms that closely follows the pattern of parametrized diamonds, studied by Moore, Hrušák, and Džamonja in [13]. However, our approach appeals to model theoretic / forcing theoretic notions, rather than pure combinatorics. The main goal of the paper is to exhibit a surprising, close connection between seemingly very distinct principles. As an application, we show that forcing with a measure algebra preserves (a variant of) $\diamondsuit(\mathfrak{d})$, improving an old result of M. Hrušák.
We study uncountable structures similar to the Fraïssé limits. The standard inductive arguments from the Fraïssé theory are replaced by forcing, so the structures we obtain are highly sensitive to the universe of set theory. In particular, the generic structures we investigate exist only in generic extensions of the universe. We prove that in most of the interesting cases the uncountable generic structures are rigid. Moreover, we provide a (consistent) example of an uncountable, dense set of reals with the group of integers as its automorphism group.
Ancel, Dobrowolski, and Grabowski (Studia Math., 1994) proved that every countable discrete subgroup of the additive group of a normed space is free Abelian, hence isomorphic to the direct sum of a certain number of copies of the additive group of the integers. In the present paper, we take a set-theoretic approach based on the theory of elementary submodels and the Singular Compactness Theorem to remove the cardinality constraint from their result and prove that indeed every discrete subgroup of the additive group of a normed space is free Abelian.
We introduce a new weak variation of diamond that is meant to only guess the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by proving it is compatible with Martin's axiom. We then prove that this principle is nontrivial by showing it may consistently fail.
We introduce the family of axioms, denoted $\operatorname{Slice}_κ$, that claim the existence of strictly increasing decompositions of the form $$2^δ=\bigcup_{α<κ} 2^δ\cap M_α,$$ where $δ<κ$, and $\{M_α|\; α<κ\}$ is a $\subseteq$-increasing sequence of transitive models of set theory. We study compatibility of these axioms with versions of Martin's Axiom, and in particular show that $\operatorname{Slice}$ is compatible only with some very weak form of $MA$.
Building on the work of Avraham, Rubin, and Shelah, we aim to build a variant of the Fraïssé theory for uncountable models built from finite submodels. With this aim, we generalize the notion of an increasing set of reals to other structures. As an application, we prove that the following is consistent: there exists an uncountable, separable metric space $X$ with rational distances, such that every uncountable partial 1-1 function from $X$ to $X$ is an isometry on an uncountable subset. We aim for a general theory of structures with this kind of properties. This includes results about the automorphism groups, and partial classification results.
The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality $\mathfrak{c}$. We provide some examples of pairwise non-isomorphic countably saturated linear orders of cardinality $\mathfrak{c}$, under different set-theoretic assumptions. We give a new proof of the old theorem of Harzheim, that the class of countably saturated linear orders has a uniquely determined one-element basis. From our proof it follows that this minimal linear order is a Fraïsse limit of certain Fraïsse class. In particular, it is homogeneous with respect to countable subsets. Next, we prove the existence and uniqueness of the uncountable version of the random graph. This graph is isomorphic to $(H(ω_1),\in \cup \ni)$, where $H(ω_1)$ is the set of hereditarily countable sets, and two sets are connected if one of them is an element of the other. In the last section, an example of a prime countably saturated Boolean algebra is presented.
Let $(X, +)$ denote $(\mathbb{R}, +)$ or $(2^ω, +_2)$. We prove that for any meagre set $F \subseteq X$ there exists a subgroup $G \le X$ without the Baire property, disjoint with some translation of F. We point out several consequences of this fact and indicate why analogous result for the measure cannot be established in ZFC. We extend proof techniques from the work of Rosłanowski and Shelah [1].